The speed \(v\)  as function of \(\vartheta \)  is given in (REF ). We now invert this function, i.e. we solve for \(\vartheta \) .
This can be done directly using elliptic functions Some basic knowledge of Elliptic functions is assumed, see e.g. the book
of Whittacker and Watson [1]}. For the definition of the functions sn, cn, and dn see appendix ..
We set \(\vartheta = \rm {am}(u,k)\) , where \(\rm {am}(u,k)\)  is Jacobi's amplitude function, which has as input \(u\)  and the modulus \(k\)  with \(k=e\)  and get
\(v = \frac{2 \pi a}{T} \frac{1}{\sqrt{1-e^2}} \, \left( \rm {dn}(u,e) + e \cdot \rm {cn}(u,e) \right) \mbox{ . }\) 
